Learning-Augmented FEM–BEM Solvers for Biomolecular Simulations

About the Project

Electrostatic interactions play a central role in biomolecular processes, including protein–ligand binding, RNA folding, and vaccine design. The Poisson–Boltzmann equation (PBE), particularly in its non-linear form, is a foundational tool in modelling these interactions. Despite its theoretical power, practical application of the PBE remains limited by computational constraints—especially for complex biomolecular geometries and high-charge regimes where traditional linearised solvers fail to deliver physical fidelity.

Recent work has demonstrated that coupling the finite element method (FEM) and boundary element method (BEM) can significantly improve the stability and accuracy of PBE solvers. In particular, I have shown that FEM–BEM coupling allows volumetric flexibility (FEM) and precise boundary representation (BEM) to work in tandem, yielding accurate solvation free energy estimates for challenging molecular systems [1]. However, as the complexity of target systems increases, challenges persist, particularly around solver convergence, preconditioning, and adaptivity to system-specific features.

In parallel, machine learning (ML) has emerged as a promising tool to augment PDE solvers. Recent advances in physics-informed neural networks (PINNs) have enabled direct learning of electrostatic potentials from geometries and charges [2], bypassing the need for full mesh-based discretisation. While these approaches show potential, they currently struggle with high-dimensional problems, scaling to large molecular systems, and enforcing the precise physics required in charged solvation environments.

This project proposes a new research direction: a learning-augmented FEM–BEM solver for the Poisson–Boltzmann equation that combines the rigour of traditional numerical methods with the efficiency and adaptivity of modern ML techniques.

Funding Notes

there is no funding for this project

References

1. M. Bosy, M. W. Scroggs, T. Betcke, E. Burman, C. D. Cooper, J. Comput. Chem., 2024, 45(11), 787.
2. Achondo, M.A., Chaudhry, J.H. and Cooper, C.D., 2025. An Investigation of Physics Informed Neural Networks to Solve the Poisson–Boltzmann Equation in Molecular Electrostatics. Journal of Chemical Theory and Computation, 21(7), pp.3726–3744.